Analysis and Plotting of Time Series in R

0.1 Introduction

As discussed, a time series can be broken down to its components so as to systematically understand, analyze, model and forecast it. We have to begin by answering fundamental questions such as:

1. What are the types of time series?
2. How does one process and analyze time series data?
3. How does one plot time series?
4. How to decompose it? How to extract a level, a trend, and seasonal components from a time series?
5. What is auto correlation etc.

What is a stationary time series?

0.2 Case Study -1: Walmart Sales Dataset from timetk

Let us inspect what datasets are available in the package timetk. Type data(package = "timetk") in your Console to see what datasets are available.

Let us choose the Walmart Sales dataset. See here for more details: Walmart Recruiting - Store Sales Forecasting |Kaggle

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data("walmart_sales_weekly")
walmart_sales_weekly

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inspect(walmart_sales_weekly)

categorical variables:  
       name     class levels    n missing
1        id    factor   3331 1001       0
2 IsHoliday   logical      2 1001       0
3      Type character      1 1001       0
                                   distribution
1 1_1 (14.3%), 1_3 (14.3%), 1_8 (14.3%) ...    
2 FALSE (93%), TRUE (7%)                       
3 A (100%)                                     

Date variables:  
  name class      first       last min_diff max_diff    n missing
1 Date  Date 2010-02-05 2012-10-26   0 days   7 days 1001       0

quantitative variables:  
           name   class         min          Q1      median          Q3
1         Store numeric      1.0000      1.0000      1.0000      1.0000
2          Dept numeric      1.0000      3.0000     13.0000     93.0000
3  Weekly_Sales numeric   6165.7300  28257.3000  39886.0600  77943.5700
4          Size numeric 151315.0000 151315.0000 151315.0000 151315.0000
5   Temperature numeric     35.4000     57.7900     69.6400     80.4900
6    Fuel_Price numeric      2.5140      2.7590      3.2900      3.5940
7     MarkDown1 numeric    410.3100   4039.3900   6154.1400  10121.9700
8     MarkDown2 numeric      0.5000     40.4800    144.8700   1569.0000
9     MarkDown3 numeric      0.2500      6.0000     25.9650    101.6400
10    MarkDown4 numeric      8.0000    577.1400   1822.5500   3750.5900
11    MarkDown5 numeric    554.9200   3127.8800   4325.1900   6222.2500
12          CPI numeric    210.3374    211.5312    215.4599    220.6369
13 Unemployment numeric      6.5730      7.3480      7.7870      7.8380
           max         mean           sd    n missing
1       1.0000 1.000000e+00 0.000000e+00 1001       0
2      95.0000 3.585714e+01 3.849159e+01 1001       0
3  148798.0500 5.464634e+04 3.627627e+04 1001       0
4  151315.0000 1.513150e+05 0.000000e+00 1001       0
5      91.6500 6.830678e+01 1.420767e+01 1001       0
6       3.9070 3.219699e+00 4.260286e-01 1001       0
7   34577.0600 8.090766e+03 6.550983e+03  357     644
8   46011.3800 2.941315e+03 7.873661e+03  294     707
9   55805.5100 1.225400e+03 7.811934e+03  350     651
10  32403.8700 3.746085e+03 5.948867e+03  357     644
11  20475.3200 5.018655e+03 3.254071e+03  357     644
12    223.4443 2.159969e+02 4.337818e+00 1001       0
13      8.1060 7.610420e+00 3.825958e-01 1001       0

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glimpse(walmart_sales_weekly)

Rows: 1,001
Columns: 17
$ id           <fct> 1_1, 1_1, 1_1, 1_1, 1_1, 1_1, 1_1, 1_1, 1_1, 1_1, 1_1, 1_…
$ Store        <dbl> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, …
$ Dept         <dbl> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, …
$ Date         <date> 2010-02-05, 2010-02-12, 2010-02-19, 2010-02-26, 2010-03-…
$ Weekly_Sales <dbl> 24924.50, 46039.49, 41595.55, 19403.54, 21827.90, 21043.3…
$ IsHoliday    <lgl> FALSE, TRUE, FALSE, FALSE, FALSE, FALSE, FALSE, FALSE, FA…
$ Type         <chr> "A", "A", "A", "A", "A", "A", "A", "A", "A", "A", "A", "A…
$ Size         <dbl> 151315, 151315, 151315, 151315, 151315, 151315, 151315, 1…
$ Temperature  <dbl> 42.31, 38.51, 39.93, 46.63, 46.50, 57.79, 54.58, 51.45, 6…
$ Fuel_Price   <dbl> 2.572, 2.548, 2.514, 2.561, 2.625, 2.667, 2.720, 2.732, 2…
$ MarkDown1    <dbl> NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, N…
$ MarkDown2    <dbl> NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, N…
$ MarkDown3    <dbl> NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, N…
$ MarkDown4    <dbl> NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, N…
$ MarkDown5    <dbl> NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, N…
$ CPI          <dbl> 211.0964, 211.2422, 211.2891, 211.3196, 211.3501, 211.380…
$ Unemployment <dbl> 8.106, 8.106, 8.106, 8.106, 8.106, 8.106, 8.106, 8.106, 7…

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# Try this in your Console
# help("walmart_sales_weekly")

The data is described as:

A tibble: 9,743 x 3

• id Factor. Unique series identifier (4 total)
• Store Numeric. Store ID.
• Dept Numeric. Department ID.
• Date Date. Weekly timestamp.
• Weekly_Sales Numeric. Sales for the given department in the given store.
• IsHoliday Logical. Whether the week is a “special” holiday for the store.
• Type Character. Type identifier of the store.
• Size Numeric. Store square-footage
• Temperature Numeric. Average temperature in the region.
• Fuel_Price Numeric. Cost of fuel in the region.
• MarkDown1, MarkDown2, MarkDown3, MarkDown4, MarkDown5 Numeric. Anonymized data related to promotional markdowns that Walmart is running. MarkDown data is only available after Nov 2011, and is not available for all stores all the time. Any missing value is marked with an NA.
• CPI Numeric. The consumer price index.
• Unemployment Numeric. The unemployment rate in the region.

Very cool to know that mosaic::inspect() identifies date variables separately!

NOTE: 1. This is still a data.frame, with a time-oriented variable of course, and not yet a time-series object. Note that this data frame has the YMD columns repeated for each Dept. 2. The Date column has repeated entries for each Dept! To deal with this repetition, we will always need to split the Weekly_Sales by the Dept column before we plot or analyze.

Since our sales are weekly, we will convert Date to yearweek format:

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walmart_time <- walmart_sales_weekly %>% 
  # mutate(Date = as.Date(Date)) %>% 
  as_tsibble(index = Date, # Time Variable
             key = Dept 
             
  #  Identifies unique "subject" who are measures
  #  All other variables such as Weekly_sales become "measured variable"
  #  Each observation should be uniquely identified by index and key

             )
walmart_time

0.2.1 Basic Time Series Plots

The easiest way is to use autoplot from the feasts package. You may need to specify the actual measured variable, if there is more than one numerical column:

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autoplot(walmart_time,.vars = Weekly_Sales)

timetk gives us interactive plots that may be more evocative than the static plot above. The basic plot function with timetk is plot_time_series. There are arguments for the date variable, the value you want to plot, colours, groupings etc.

Let us explore this dataset using timetk, using our trusted method of asking Questions:

Q.1 How are the weekly sales different for each Department?

There are

number of Departments. So we should be fine plotting them and also facetting with them, as we will see in a bit:

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walmart_time %>%   timetk::plot_time_series(Date, Weekly_Sales,
                   .color_var = Dept, .legend_show = TRUE,
                   .title = "Walmart Sales Data by Department",
                   .smooth = FALSE)

Q.2. What do the sales per Dept look like during the month of December (Christmas time) in 2012? Show the individual Depts as facets.

We can of course zoom into the interactive plot above, but if we were to plot it anyway:

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# Only include rows from  1 to December 31, 2011
# Data goes only up to Oct 2012

walmart_time %>% 
  # Each side of the time_formula is specified as the character 'YYYY-MM-DD HH:MM:SS',
  timetk::filter_by_time(.date_var = Date,
                         .start_date = "2011-12-01",
                         .end_date = "2011-12-31") %>%

  plot_time_series(.date_var = Date, 
                   .value = Weekly_Sales, 
                   .color_var = Dept, 
                   .facet_vars = Dept, 
                   .facet_ncol = 2,
                   .smooth = FALSE) # Only 4 points per graph

Clearly the “unfortunate” Dept#13 has seen something of a Christmas drop in sales, as has Dept#38 ! The rest, all is well, it seems…

0.2.2 Too much noise? How about some averaging?

Q.3 How do we smooth out some of the variations in the time series to be able to understand it better?

Sometimes there is too much noise in the time series observations and we want to take what is called a rolling average. For this we will use the function timetk::slidify to create an averaging function of our choice, and then apply it to the time series using regular dplyr::mutate

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# Let's take the average of Sales for each month in each Department.
# Our **function** will be named "rolling_avg_month": 

rolling_avg_month = slidify(.period = 4, # every 4 weeks
                            .f = mean, # The funtion to average
                            .align = "center", # Aligned with middle of month
                            .partial = TRUE) # TO catch any leftover half weeks
rolling_avg_month

function (...) 
{
    slider_2(..., .slider_fun = slider::pslide, .f = .f, .period = .period, 
        .align = .align, .partial = .partial, .unlist = .unlist)
}
<bytecode: 0x0000027a0b5bfb28>
<environment: 0x0000027a0b5bf1f8>

OK, slidify creates a function! Let’s apply it to the Walmart Sales time series…

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walmart_time %>% 
  # group_by(Dept) %>% 
  mutate(avg_monthly_sales = rolling_avg_month(Weekly_Sales)) %>% 
  # ungroup() %>% 
  timetk::plot_time_series(Date, avg_monthly_sales,.color_var = Dept, .smooth = FALSE)

Curves are smoother now. Need to check whether the averaging was done on a per-Dept basis…should we have had a group_by(Dept) before the averaging, and ungroup() before plotting? Try it !!

0.2.3 Decomposing Time Series: Trends, Seasonal Patterns, and Cycles

Each data point (\(Y_t\)) at time \(t\) in a Time Series can be expressed as either a sum or a product of 4 components, namely, Seasonality(\(S_t\)), Trend(\(T_t\)), Cyclic, and Error(\(e_t\)) (a.k.a White Noise).

• Trend: pattern exists when there is a long-term increase or decrease in the data.
• Seasonal: pattern exists when a series is influenced by seasonal factors (e.g., the quarter of the year, the month, or day of the week).
• Cyclic: pattern exists when data exhibit rises and falls that are not of fixed period (duration usually of at least 2 years).
• Error or Noise: Random component

Decomposing non-seasonal datas means breaking it up into trend and irregular components.To estimate the trend component of a non-seasonal time series that can be described using an additive model, it is common to use a smoothing method, such as calculating the simple moving average of the time series.

timetk has the ability to achieve this: Let us plot the trend, seasonal, cyclic and irregular aspects of Weekly_Sales for Dept 38:

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walmart_time %>% 
  filter(Dept == "38") %>% 
  timetk::plot_stl_diagnostics(.date_var = Date, .value = Weekly_Sales)

frequency = 13 observations per 1 quarter

trend = 52 observations per 1 year

We can do this for all Dept using fable and fabletools:

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walmart_decomposed <- 
  walmart_time %>% 
  
  # If we want to filter, we do it here
  # filter(Dept == "38") %>% 
  # 

fabletools::model(stl = STL(Weekly_Sales))

fabletools::components(walmart_decomposed)

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autoplot(components((walmart_decomposed)))

0.3 Case Study-2: Dataset from nycflights13

Let us try the flights dataset from the package nycflights13. Try data(package = "nycflights13") in your Console.

We have the following datasets innycflights13:

Data sets in package nycflights13:

• airlines Airline names.
• airports Airport metadata
• flights Flights data
• planes Plane metadata.
• weather Hourly weather data

Let us analyze the flights data:

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data("flights", package = "nycflights13")
mosaic::inspect(flights)

categorical variables:  
     name     class levels      n missing
1 carrier character     16 336776       0
2 tailnum character   4043 334264    2512
3  origin character      3 336776       0
4    dest character    105 336776       0
                                   distribution
1 UA (17.4%), B6 (16.2%), EV (16.1%) ...       
2 N725MQ (0.2%), N722MQ (0.2%) ...             
3 EWR (35.9%), JFK (33%), LGA (31.1%)          
4 ORD (5.1%), ATL (5.1%), LAX (4.8%) ...       

quantitative variables:  
             name   class  min   Q1 median   Q3  max        mean          sd
1            year integer 2013 2013   2013 2013 2013 2013.000000    0.000000
2           month integer    1    4      7   10   12    6.548510    3.414457
3             day integer    1    8     16   23   31   15.710787    8.768607
4        dep_time integer    1  907   1401 1744 2400 1349.109947  488.281791
5  sched_dep_time integer  106  906   1359 1729 2359 1344.254840  467.335756
6       dep_delay numeric  -43   -5     -2   11 1301   12.639070   40.210061
7        arr_time integer    1 1104   1535 1940 2400 1502.054999  533.264132
8  sched_arr_time integer    1 1124   1556 1945 2359 1536.380220  497.457142
9       arr_delay numeric  -86  -17     -5   14 1272    6.895377   44.633292
10         flight integer    1  553   1496 3465 8500 1971.923620 1632.471938
11       air_time numeric   20   82    129  192  695  150.686460   93.688305
12       distance numeric   17  502    872 1389 4983 1039.912604  733.233033
13           hour numeric    1    9     13   17   23   13.180247    4.661316
14         minute numeric    0    8     29   44   59   26.230100   19.300846
        n missing
1  336776       0
2  336776       0
3  336776       0
4  328521    8255
5  336776       0
6  328521    8255
7  328063    8713
8  336776       0
9  327346    9430
10 336776       0
11 327346    9430
12 336776       0
13 336776       0
14 336776       0

time variables:  
       name   class               first                last min_diff   max_diff
1 time_hour POSIXct 2013-01-01 05:00:00 2013-12-31 23:00:00   0 secs 25200 secs
       n missing
1 336776       0

We have time-related columns; Apart from year, month, day we have time_hour; and time-event numerical data such as arr_delay (arrival delay) and dep_delay (departure delay). We also have categorical data such as carrier, origin, dest, flight and tailnum of the aircraft. It is also a large dataset containing 330K entries. Enough to play with!!

Let us replace the NAs in arr_delay and dep_delay with zeroes for now, and convert it into a time-series object with tsibble:

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flights_delay_ts <- flights %>% 
  
  mutate(arr_delay = replace_na(arr_delay, 0), 
         dep_delay = replace_na(dep_delay, 0)) %>% 
  
  select(time_hour, arr_delay, dep_delay, carrier, origin, dest, flight, tailnum) %>% 
  tsibble::as_tsibble(index = time_hour, 
                      # All the remaining identify unique entries
                      # Along with index
                      # Many of these variables are common
                      # Need *all* to make unique entries!
                      key = c(carrier, origin, dest,flight, tailnum), 
                      validate = TRUE) # Making sure each entry is unique


flights_delay_ts

Q.1. Plot the monthly average arrival delay by carrier

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mean_arr_delays_by_carrier <- 
  flights_delay_ts %>%
  group_by(carrier) %>% 
  
  index_by(month = ~ yearmonth(.)) %>% 
  # index_by uses (year, yearquarter, yearmonth, yearweek, as.Date)
  # to create a new column to show the time-grouping
  # year / quarter / month/ week, or day...
  # which IS different from traditional dplyr
  
  summarise(mean_arr_delay = mean(arr_delay, na.rm = TRUE)
  )

mean_arr_delays_by_carrier

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mean_arr_delays_by_carrier %>%
  timetk::plot_time_series(
    .date_var = month,
    .value = mean_arr_delay,
    .facet_vars = carrier,
    .smooth = FALSE,
    # .smooth_degree = 1,
    
    # keep .smooth off since it throws warnings if there are too few points
    # Like if we do quarterly or even yearly summaries
    # Use only for smaller values of .smooth_degree (0,1)
    #
    .facet_ncol = 4,
    .title = "Average Monthly Arrival Delays by Carrier"
  )

Q.2. Plot a candlestick chart for total flight delays by month for each carrier

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flights_delay_ts %>% 
  mutate(total_delay = arr_delay + dep_delay) %>%
  timetk::plot_time_series_boxplot(
    .date_var = time_hour,
    .value = total_delay,
    .color_var = origin,
    .facet_vars = origin,
    .period = "month",
  # same warning again
    .smooth = FALSE
  )

Q.2. Plot a heatmap chart for total flight delays by origin, aggregated by month

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avg_delays_month <- flights_delay_ts %>% 
  group_by(origin) %>% 
  mutate(total_delay = arr_delay + dep_delay) %>% 
  index_by(month = ~ yearmonth(.)) %>% 
  # index_by uses (year, yearquarter, yearmonth, yearweek, as.Date)
  # to create a new column to show the time-grouping
  # year / quarter / month/ week, or day...
  # which IS different from traditional dplyr
    summarise(mean_monthly_delay = mean(total_delay, na.rm = TRUE)
  )

avg_delays_month 

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# three origins 12 months therefore 36 rows
# Tsibble index_by + summarise also gives us a  month` column 



ggformula::gf_tile(origin ~ month, fill = ~ mean_monthly_delay, 
                   color = "black", data = avg_delays_month,
                   title = "Mean Flight Delays from NY Airports in 2013") %>% 
  gf_theme(theme_classic()) %>% 
  gf_theme(scale_fill_viridis_c(option = "A")) %>% 
  
# "magma" (or "A") inferno" (or "B") "plasma" (or "C") 
# "viridis" (or "D") "cividis" (or "E") 
# "rocket" (or "F") "mako" (or "G") "turbo" (or "H")

  gf_theme(scale_x_time(breaks = 1:12, labels = month.abb))

1 Conclusion

We can plot most of the common Time Series Plots with the help of the tidyverts packages: ( tsibble, feast, fable and fabletools) , along with timetk and ggformula.

There are other plot packages to investigate, such as dygraphs

Recall that we have used the tsibble format for the data. There are other formats such as ts, xts and others which are meant for time series analysis. But for our present purposes, we are able to do things with the capabilities of timetk.

2 References

1. Rob J Hyndman and George Athanasopoulos, Forecasting: Principles and Practice (3rd ed), Available Online https://otexts.com/fpp3/